To construct the ROM basis, we first collect snapshots from the simulation of the FOM.
If we are interested in time prediction (as in the numerical illustration in Section ), the snapshots can be FEM approximations of (REF ) at the time instances \(t_{1}, \ldots , t_{M}\) , i.e., \({u}_{h}^{1}, \ldots , {u}_{h}^{M}\) , respectively.
(If (REF ) depends on parameters, we can also build a ROM basis for parameter prediction [1]}, [2]}.)
Next, we use these snapshots to construct the modes \(\lbrace  {\varphi }_1, \ldots , {\varphi }_R \rbrace \) , which represent the recurrent spatial structures in the system described by (REF ).
Different approaches can be used to construct the ROM basis functions, e.g.,
(i) the proper orthogonal decomposition (POD) [3]}, [4]}, [5]}, [6]}, [7]};
(ii) the reduced basis method (RBM) [1]}, [2]};
(iii) the proper generalized decomposition (PGD) [10]};
and
(iv) clustering [11]}.
In this tutorial, to fix ideas, we exclusively use the POD to generate the ROM basis.
